Optimal. Leaf size=101 \[ \frac{(1-2 x)^{7/2}}{63 (3 x+2)^3}-\frac{53 (1-2 x)^{5/2}}{189 (3 x+2)^2}+\frac{265 (1-2 x)^{3/2}}{567 (3 x+2)}+\frac{530}{567} \sqrt{1-2 x}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]
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Rubi [A] time = 0.0237164, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 206} \[ \frac{(1-2 x)^{7/2}}{63 (3 x+2)^3}-\frac{53 (1-2 x)^{5/2}}{189 (3 x+2)^2}+\frac{265 (1-2 x)^{3/2}}{567 (3 x+2)}+\frac{530}{567} \sqrt{1-2 x}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx &=\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}+\frac{106}{63} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}-\frac{53 (1-2 x)^{5/2}}{189 (2+3 x)^2}-\frac{265}{189} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}-\frac{53 (1-2 x)^{5/2}}{189 (2+3 x)^2}+\frac{265 (1-2 x)^{3/2}}{567 (2+3 x)}+\frac{265}{189} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{530}{567} \sqrt{1-2 x}+\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}-\frac{53 (1-2 x)^{5/2}}{189 (2+3 x)^2}+\frac{265 (1-2 x)^{3/2}}{567 (2+3 x)}+\frac{265}{81} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{530}{567} \sqrt{1-2 x}+\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}-\frac{53 (1-2 x)^{5/2}}{189 (2+3 x)^2}+\frac{265 (1-2 x)^{3/2}}{567 (2+3 x)}-\frac{265}{81} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{530}{567} \sqrt{1-2 x}+\frac{(1-2 x)^{7/2}}{63 (2+3 x)^3}-\frac{53 (1-2 x)^{5/2}}{189 (2+3 x)^2}+\frac{265 (1-2 x)^{3/2}}{567 (2+3 x)}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0141768, size = 42, normalized size = 0.42 \[ \frac{(1-2 x)^{7/2} \left (\frac{2401}{(3 x+2)^3}-848 \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{151263} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 66, normalized size = 0.7 \begin{align*}{\frac{40}{81}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{163}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{1505}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{6125}{108}\sqrt{1-2\,x}} \right ) }-{\frac{530\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.83644, size = 136, normalized size = 1.35 \begin{align*} \frac{265}{1701} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{40}{81} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (1467 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 6020 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6125 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52736, size = 255, normalized size = 2.52 \begin{align*} \frac{265 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (1080 \, x^{3} + 3627 \, x^{2} + 2983 \, x + 713\right )} \sqrt{-2 \, x + 1}}{1701 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.65486, size = 126, normalized size = 1.25 \begin{align*} \frac{265}{1701} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{40}{81} \, \sqrt{-2 \, x + 1} + \frac{1467 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 6020 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6125 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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